How do I animate a right-angle triangle inside a circle (hypotenuse of the triangle is the radius of the circle)
Start by adding a single vertex (ShiftA then S then A or Add -> Single Vert -> Add Single Vert). You should be in edit mode by default after doing this, but if not Tab into edit mode.

Extrude by 1 along the X axis (E then X then Numpad 1), then extrude by 1 along the Y axis (E then Y then Numpad 1). Select everything, by hitting A twice. Press F to make a face.



If you want only the outline of the triangle, press I followed by some sufficiently small number then press Enter, then X followed by F. Tab out of edit mode.



Go to the object properties and add the following into the Scale boxes:

If you prefer degrees:
X: #cos(radians(360*(frame-1)/100))
Y: #sin(radians(360*(frame-1)/100))

If you prefer radians:
X: #cos(2*pi360*(frame-1)/100)
Y: #sin(2*pi*(frame-1)/100)

The hash signs are mandatory. They tell Blender that you're entering a driver rather than a static value. If you edit the driver later, the hash signs won't show up, and you shouldn't re-add them. Once the driver is created, Blender knows it's a driver without being told again.



frame is the current frame number, and the reason I subtract 1 is that Blender starts at frame 1, while the parametric equations for a circle start at 0. 360 or 2*pi is the angle of a full circle, and 100 is the number of frames it will take to complete a full circle.

The result:



As you can see, the triangle's outline gets scaled as well, and there's certainly room for improvement there, however, here's the .blend.